If α is a non-zero algebraic number over ℚ, then eα is transcendental over ℚ.

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e is transcendental over ℚ.

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π is transcendental over ℚ. As a result, it is impossible to “square the circle”!

It is easy to see that π is transcendental over ℚ⁢(e) iff e is transcendental over ℚ⁢(π) iff π and e are algebraically independent. However, whether π and e are algebraically independent is still an open question today.